Maths 1st-year complete syllabus
Ignou provided Bscg 1st year syllabus for BSCG students. Here we provide you complete Syllabus for Maths (BMTC). You can download these syllabi in pdf.
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This is a first-level course, consisting of five blocks, and is intended as a short introduction to calculus. Calculus is increasingly being recognized, and accepted, as a powerful tool in the exact sciences and social sciences. Its power stems from two sources – the derivative and the integral. In this course, we shall acquaint you with the basic techniques of differential and integral calculus. We shall also briefly trace the historical development of calculus.
We shall begin the course with essential preliminary concepts, in the first block. You will be introduced the concepts of ‘limit’, ‘continuity’, and ‘derivative’ in the second and third blocks. We shall discuss the geometrical significance and application of the derivative in the fourth block. The fifth block focuses on the other important concept of calculus, namely, integral.
Definition and examples of sets and subsets, Venn diagrams, Complementation, Intersection, Union, Distributive laws, De Morgan’s laws, Cartesian Product, Relations and Functions, Composition of Functions, and Binary Operation, Operations (inverse, composite). The cartesian system, Graphs of Functions, Equations of a line, Symmetry, change of axis, polar coordinates. Definition and examples of complex numbers, Geometric representation and polar representation, algebraic operations, De Moivre’s theorem, trigonometric identities, nth roots of a complex number. Basic Theory of Equations: Relations between roots and coefficients; Descartes rule of signs, Solution of equations up to bi-quadratic equations.
Real number line, Supremum and Infimum, Absolute value, Interval and some special types of functions (even, odd, monotonic, periodic). Definition of Limits, Algebra of limits, One-
sided limits, The concept of infinite limits (infinite limits as the independent variable
- one-sided infinite limits, limits as the independent variable tends to or - ). Continuity, algebra of continuous functions, Types of discontinuity.
Derivatives of some simple functions, algebra of derivatives, the chain rule, continuity versus derivability. Derivatives of the various trigonometric functions, the derivative of the inverse of a function. The inverse function theorem, derivatives of inverse trigonometric functions, use of transformations. The derivative of an exponential function, logarithmic functions, hyperbolic functions, inverse hyperbolic functions, methods of differentiation (derivative of xr, logarithmic differentiation, derivatives of functions defined in terms of a parameter, derivatives of implicit functions).
Higher-order derivatives: Second and third-order derivatives, nth order derivatives, Leibnitz Theorem, Taylor Polynomials. Indeterminate forms: L’Hopital’s rule for 0 forms, L’Hopital’s
the rule for form, other types of indeterminate forms (indeterminate forms of the type,
indeterminate forms of the type 0. , indeterminate forms of type 00 , 0 ,1 ) Ups and Downs: Rolle’s Theorem, Lagrange’s mean value theorem, Maxima-minima of functions (Definitions and examples, a necessary condition for the existence of extreme points, Sufficient conditions for the existence of extreme points, first derivative test, second derivative test), Monotonicity, Curvature, Tangents and Normals, Angles of the intersection of two curves, Concavity / Convexity, points of inflection. Classifying singular points, Asymptotes (Parallel to the axes, Oblique asymptotes), Tracing of the curve.
Introduction to Integration: UPF, LPF, Definite integral, properties, Fundamental theorem of calculus (without proof.), Standard integrals, Algebra of integrals. Methods of Integration, Reduction Formula. Applications of Integration: Area under a curve, area bounded by a closed curve, length of a plane curve, Volume and Surface area of a solid generated by revolution.
This course of Differential Equation assumes the knowledge of the course BMTC-131 on Calculus. The studies in this course are divided into four blocks.
Block-1 deals with functions of two and three real variables. The purpose of this block is to provide the basis for studying the remaining blocks of the course. We have given a brief discussion on 3D-cordinate system and discussed the algebraic and geometrical structure of R2 and R3. The notions of limit, continuity and differentiability are extended for functions of 2 and 3 variables. This block also covers chain rule and homogeneous functions.
We have started Block-2 with the essentials and the basic definitions related to the study of differential equations. After discussing various methods of solving first order ordinary differential equations (ODEs) we have formulated some of the problems of physical and engineering interest in terms of first order linear differential equations. In Block-3 we have laid specific stress on the applications of second order ODEs.
In Block-4 we have discussed simultaneous, total and partial differential equations (PDEs). Here we have classified the first order PDEs into linear, semi-linear, quasi-linear and non-linear PDEs and discussed the various types of solutions/integrals of these PDEs.
All the concepts discussed are followed by a lot of examples as well as exercises. These will help you get a better grasp of the techniques discussed in this course.
3D-Cartesian Coordinate System, Spherical Coordinate System and Cylindrical Coordinate System, Cartesian products, Properties of Rn (Linear Space Properties), Distance in R2 and R3 , Functions from Rn to R (n = 2, 3), Limit of functions from R2 R and from R3 R, Repeated Limits, Properties of Limits, Continuity of functions from R2 R and from
R3 R, Algebra of continuous functions. First Order Partial Derivatives, Geometrical Meaning, Continuity and Partial Derivatives, Differentiability of functions from R2 R ,
Differentiability of functions from R3 R, Higher Order Partial Derivatives, Equality of Mixed Partial Derivatives (Euler’s , Schwarz’s and Young’s Theorem without proof), Chain rule for
finding partial derivatives of composit functions, Total Derivative, Homogeneous functions and Euler’s theorem.
Basic concepts in the theory of differential equations, Family of curves and differential equations, Differential Equations arising from physical situations. Separation of Variables, Homogeneous equations, Exact equations, Integrating factors. Classification of first order differential equations (DE), General solutions of linear non-homogeneous equation, Method of Undetermind coefficient, Method of Variation of Parameters, Equations reducible to linear form, Applications of linear DEs. Equations which can be factorized, Equations which cannot be factorized (solvable for x, y, independent or dependent variable is absent, homogeneous in x and y, Clairaut’s and Riccati’s equations).
General form of linear ordinary differential equation, Condition for the existence of unique solution, linear dependence and independence of the solution of DEs, Method of solving homogeneous equation with constant coefficients; Method of undetermined coefficients – Types of non-homogeneous terms for which the method is applicable (polynomial, exponential, sinusoidal etc.), Observations and Constraints of the method. Variation of parameters, Reduction of order, Euler’s equations. Differential operators, General method of finding Particular Integral (PI), Short method of finding PI, Applications – Mechanical Vibrations, Electric Circuits.
Curves and surfaces in space, Formation of simultaneous DEs, Methods of solution – Method of Multipliers, One Variable absent, Applications – Particle motion in phase-space, Electric Circuits. Total Differential Equations – Definition and examples, Integrability condition (only statement and illustration), Methods of Integration (By Inspection, Variable separable, One variable separable, Homogeneous equation). Origin, Classification (order, degree, linear, semi-linear, non-linear) of linear first order PDEs, Formation of Linear Equations of the First Order and types of their solutions, Lagranges Method, Solutions of non-linear PDEs – The Complete integral, Compatible system of first order equations, Charpits method, Standard forms.